Sufficient conditions for the unique solvability of linear networks containing memoryless 2-ports

Combinatorial necessary and sufficient conditions for the unique solvability of linear networks containing n-ports are well known for the ‘general’ case. They are only necessary if relations among n-port parameters are also taken into consideration. In the present paper combinatorial sufficient conditions are presented for linear networks containing RLC elements and memoryless 2-ports. The somewhat surprising result is proved that whether a 2-port can cause certain types of singularities can be predicted before the interconnection. A concept, similar to the normal tree (which intersects ideal transformers by one, gyrators by two or no edges) is introduced for arbitrary 2-ports. Its existence implies unique solvability. Relations to previous results and algorithmical aspects are also discussed.     

On the minimum hybrid rank of a graph relative to a partition of its edges and its application to electrical network analysis

We consider the following problem in this paper: given a partition of the set of edges of a graph such that the subgraphs on the blocks of the partition are connected, what is the minimum number of node fission (splitting a node into two) and node fusion (coalescing two nodes) operations that will destroy all circuits containing edges from more than one block of the partition? This problem can be seen to be a generalization of the problem of determining the minimum hybrid rank of a graph. We may therefore regard this minimum number as the minimum hybrid rank of the graph relative to a specified partition of the edge set of the graph. We give an efficient algorithm for the construction of a minimum length sequence of such operations. Our solution can be applied to the problem of analysing networks after first decomposing them into subnetworks-the equations of the network can be displayed in a bordered block diagonal form where the blocks along the diagonal correspond to the subnetworks and the thickness of the border corresponds to the minimum hybrid rank relative to the partition of the sets of edges of the network into the sets of edges of the subnetworks.     

Current and voltage graphs of a composite graph

From a given composite graph two graphs are derived, which can be interpreted as current and voltage graphs, respectively. It is shown that a complete tree of both graphs is a directed tree of the composite graph. Thus the generation of directed trees is reduced to the generation of complete trees. The matrix belonging to the composite graph is decomposed in the incidence matrices of the current and voltage graphs and in a diagonal matrix whose elements are given by the edge-weights of the composite graph. The number of directed trees in a composite graph is determined using the incidence matrices of both graphs.     

Initial vertex partitioning and testing isomorphism of graphs and digraphs

In this paper, we present a new method for initial vertex partitioning of graphs (digraphs) using the notion of a vertex label that includes several parameters of the graph for each vertex of the graph (digraph). Our method provides a more refined initial vertex partitioning for graphs (digraphs) than any of the known methods in O(nE) time for a graph with n vertices and E edges. Several examples are given as illustrations of the method. A computational comparison of our method with the Schmidt-Druffel algorithm is presented. The method can be coupled with any existing good back-tracking method for testing isomorphism of graphs and digraphs.     

Some topologico-dynamical properties of linear passive reciprocal networks

The paper considers the complete solvability and the order of complexity of passive RLCT (T = multiwinding ideal transformer) networks. A topological approach based on the determinant polynomial of the matrix of hybrid equations, formed as a set of 1st-order differential and algebraic equations, reveals the structure of the formulation tree and the subnetworks accountable for degeneracies. Topological and algebraic degeneracies are defined. Necessary and sufficient conditions for complete solvability are derived, and two algorithms are given to determine the order of complexity topologically, i.e. without having an explicit state-space representation.     

Platonic Body Resistance Networks

There are surprisingly few graphs which are both regular (same degree of incidence at each node) and have regular duals. Because these graphs all can be arranged to appear as regular geometric solids, they are known, specifically, as Platonic bodies. While direct application of graph theoretical methods can be used to discover all the graphs, this paper shows how topological and electrical restrictions also can be used to find the Platonic bodies through graphs of resultant resistance networks.     

Extension of the normal tree method

The results of this paper are applicable to linear electrical networks that may contain ideal transformers, nullors, independent and controlled sources, resistors, inductors, and capacitors, and, under a topological restriction, gyrators. A relation between summands of some expansion of the network determinant and pairs of conjugate trees is proved, which uncovers the equivalence of known criteria on generic solvability based on matroids and those based on pairs of conjugate trees. New criteria on the solvability of active networks are given. A method to obtain complete sets of generic state co‐ordinates is established, which includes the following extension of the wellknown normal tree method: The generic order of complexity equals the sum of the number of forest capacitors and the number of co–forest inductors in any normal pair of conjugate trees, the latter term being introduced in this paper. The voltages across the forest capacitors together with the currents through the co‐forest inductors may be given initial values independently from each other. Further, a systematic method of augmentation that yields networks of generic index 1 is proposed. All results are expressed in terms of network determinants as well as in terms of network graphs, and all given criteria may be checked by efficient algorithms. Copyright © 1999 John Wiley & Sons, Ltd.     

A graph theoretical interpretation of nonsymmetric permutation on sparse matrices

In the tableau approach to large-electrical-network analysis, as well as in structure analysis, the finite-element method, linear programming etc., a very sparse linear algebraic set of equations Ax = b has to be solved repeatedly. To efficiently solve the system via Gaussian elimination, an optimization problem has to be faced: the selection of a pivot strategy to maintain the sparsity of the matrix A . It is possible also to follow a different strategy to fully exploit the sparsity of A , i.e. to transform A into an equivalent but more convenient form. Both of these problems have been studied and partially solved by means of directed graphs associated with A when symmetric permutations on A are allowed. In this paper, a graph theoretical interpretation has been given to nonsymmetric permutations on A , which can be considered a fundamental step towards the solution of the above-mentioned optimization problems. This interpretation is obtained through decomposition theorems on nonsymmetric permutations, correspondence theorems between column (row) permutations and topological operations on a directed graph representing A .     

Applications of Linear Graph Theory to Lumped Models of Solid-State Devices

Linear graph theory has received wide application in electrical network theory. This paper extends the well-known methods of network analysis by exploiting the linear relationship between current and carrier density that exists in the lumped (large-signal) model of the bipolar transistor. The lumped model is described briefly and a linear graph is drawn for the lumped model. Relations between the trees and two-trees of this graph and the incrementals ?, Ico, and Ieo are then derived. A lumped model for a double emitter transistor is also considered and its graph is drawn. Linear graphs are then used to determine the lumped parameters of a and reverse currents for two conditions: 1) emitter one as the input and emitter two open circuited and 2) emitter one as the input and emitter two shorted. Each condition leads to its own graph which may then be used to find the applicable a and reverse currents. Methods for determining the circuit parameters are fully outlined and some sample calculations are given.     

基于GN分区的输电断面快速搜索

提出了一种基于电网分区的输电断面快速搜索方法。该方法在利用复杂网络社团划分中的GN算法对电网进行分区的基础上,结合图论中的路径矩阵等相关知识实现输电断面搜索。同时在对电网进行分区时,本文通过关键介数等指标确保脆弱线路以及重载线路优先成为区间联络线,从而有效克服了传统电网分区由于忽略电网拓扑结构以及线路负载情况而存在的问题,以此最大程度减小未进行区内选择带来的漏选问题。采用IEEE39节点系统标准算例,验证所提方法有效可行。     

Uncountable transfinite graphs and electrical networks

Up to now the theory of infinite and transfinite graphs and electrical networks has been established only for the case where the branch sets are countable. This work extends that prior theory to graphs and networks with uncountable branch sets, a structure that arises quite naturally in the theory of transfinite graphs. Moreover, conditions on the local structure of the transfinite graph are given which insure the countability of the branch set. Finally, it is shown that only countably many of the branches in an uncountable electrical network can have non-zero voltages and currents, and therefore the restriction to countability in the prior theory is not essentially restrictive.     

Hilbert Graph: An Expandable Interconnection for Clusters

This paper presents a new interconnection network inspired by chordal graphs and the small-world phenomenon, with an emphasis in providing an incrementally expandable network with a fixed degree and a small diameter. Whereas most small-world graphs are constructed by a random process, the proposed graphs are completely deterministic and regular. The Hilbert graph is constructed by a number of nodes laid down in a Hilbert curve to which a number of additional edges forming an extended mesh are incorporated. It is shown that the resulting structures are quasi planar, and posses the same cutwidth complexity of a two dimensional torus. Furthermore the Hilbert graph posses the same fixed degree and incrementally expandable properties of the torus and mesh; however it is shown that the Hilbert graph has a much larger tolerance to an increase in traffic, and can therefore be expanded to much larger systems than the mesh or torus. In particular it is shown that with an equal congestion, the number of nodes on a Hilbert network is at least O(N^2.477) larger than that of a torus. Furthermore, while broadcast in the two dimensional torus takes at least N^1/2 steps, it is shown that the same operation can be accommodated in O(N^1/4) steps in a Hilbert graph.     

The topological foundations of electrical networks — complete models

The aim of this paper is to identify the topological entities underlying the construction of a complete (not reduced) network model. Consider an active network as a pair of physically different structures, i.e. a ‘dead network’ and an ‘excitation’. First, we define the general structure of the excitation which can be applied to a dead network; then the graphs G_a and G of the active and dead networks, respectively, are introduced. It is shown that, if a complete network model is to be constructed, the topological relations among the voltages and those among the currents of the network elements must be written choosing maximal independent sets of loops and cut-sets of G_a. It is successively pointed out that the graph G can be obtained from G_a by means of a reduction operation, and the topological entities of G corresponding to the loops and to the cut-sets of G_a are singled out. It has been found that a complete model of an electric network implies the use of four topological entities defined on the set of G-edges, i.e. the closed path, the open path, the cut-set and the pseudo-cut, together with two topological entities defined on a maximal set of independent vertex couples of G, i.e. the ‘junction pair’ extreme of an open path and the ‘family of independent node couples’ split by a cut-set. Furthermore the different reference frames and the relative topological matrix transformations which allow a complete model of the network to be built up are singled out and discussed.     

结合方案的图表示

图论与结合方案有着某种共性。图论研究点与线之间的关系，结合方案则是处理之间的关系，将图论中的点看作处理，将线看作结合关系，则有些结合方案可以用图表示出来，该文对此作了些尝试，揭示了它们之间的关系。     

Global unique solvability for memristive circuit DAEs of Index 1

Known solvability results for nonlinear index‐1 differential‐algebraic equations (DAEs) are in general local and rely on the Implicit Function Theorem. In this paper, we derive a global result which guarantees unique solvability on a given time interval for a certain class of index‐1 DAEs with certain monotonicity conditions. Based on this result, we show that memristive circuit DAEs arising from the modified nodal analysis are uniquely solvable if they fulfill certain passivity and network topological conditions. Furthermore we present an error estimation for the solution with respect to perturbations on the right‐hand side and in the initial value. Copyright © 2013 John Wiley & Sons, Ltd.     

Variable dimension algorithms for solving resistive circuits

Variable dimension algorithms are a recent class of globally convergent algorithms for solving sets of non-linear equations. This paper introduces a simple circuit-theoretical interpretation of the underlying idea and describes its advantages as compared to the globally convergent methods better known in circuit theory, such as the generalized Katzenelson algorithm. the convergence criterion is also stated in topological terms, and these conditions turn out to coincide with very general sufficient conditions for existence of solutions. the approach taken here can therefore be seen as a constructive way to prove solvability of resistive circuits. the paper also discusses how to implement the algorithm in an efficient and practical way.     

基于电气耦合路径分析的割支路和割节点辨识算法

割支路和割节点的断开会导致网络产生分离子图，这种特殊的支路和节点往往代表了网络中的薄弱环节或某种优化的分割降维策略，因而在潮流、电压稳定、继电保护整定等电力系统网络计算领域具有重要的意义。以往的割支路和割节点辨识算法大多从几何拓扑的角度，将其转化为局部支路开断后的图的连通性判断。基于阻抗矩阵元素网络等值 后的节点电气耦合路径分析，提出一种识别割节点和割支路的物理拓扑判断算法。该算法利用节点阻抗矩阵反映的物理特性，引入了2个简单的数学等式判据。相对传统的图论拓扑分析算法，文中提出的算法简单、直观，具有明确的物理意义，对于理解和研究电力系统网络拓扑结构具有一定借鉴意义。     

一种生成有向树组的拓扑方法—广义树法

本文在对“王氏代数”、“k-树组”和“MINTY”这三种求有向树组的典型方法进行分析之后,提出了一种生成有向树组的拓扑方法。这种方法吸收了上述三种方法的优点,即“MINTY”的深度优先,“王氏代数”在经改良后的彻底地消除非树组合类冗余项的能力及“k-树组法”的消除对消类冗余项的能力,且由于方法本身的结构特点,增加了消除由变压器等元件的拓扑结构引起的冗余项的能力,由于引入了广义顶点的概念和能成批的生成有向树,该方法的计算机算法并没有其它算法在引入有向图时的退化现象。     

基于广义节点的配电网区域控制划分

由于配电网节点和线路数多,所以为了更好地实现配电网集中式差动保护,有必要将电网进行分区控制,以适应集控中心对信息采集和处理的要求。将配电网转化成带权的广义节点的网络,然后从图论的角度对配电网进行抽象,根据各广义节点间的拓扑结构来进行分区。通过利用"吸收法"对电网抽象图模型进行不断的简化和解环处理,最终将复杂电网分割成若干个广义节点的集合,即分割成若干个控制区域。